= Solution
Apply part (c) to the commutative unital <C-star algebra> $A=C^*(I,T)$, commutative because $T$ is normal. A character is determined by $z=\chi(T)$, since $T,T^*$ generate the algebra and $\chi(T^*)=\overline z$. The spectrum identity in part (c) shows that every $z\in\sigma_A(T)$ occurs. Thus $X\to\sigma_A(T)$ is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. Spectral permanence identifies this spectrum with the operator spectrum.
For any complex polynomial $p$ in two commuting variables, evaluation gives $\chi(p(T,T^*))=p(z,\overline z)$. Again using the spectrum identity and spectral permanence,
$$
\boxed{\sigma_{B(H)}(p(T,T^*))=\{p(z,\overline z):z\in\sigma_{B(H)}(T)\}.}
$$
This proves the requested <spectral mapping theorem>, including constants, and explains why conjugation in the second coordinate is required.
Back to article page
